Place a large, long handled skillet over medium heat. Add butter and allow to melt. When butter is completely melted, add both sugars and increase heat to medium high, stirring with a wooden spoon or heatproof spatula to combine. When sugar mixture begins to bubble and thicken, add peaches and stir or toss to coat. Allow peaches to simmer in hot caramel until they begin to soften and release their juices, three to four minutes (the peaches will start to look shiny and the liquid will begin turning a light rosy hue from the peach skins). Add the blackberries and allspice and stir to coat.
Pour the bourbon into the pan to cover the entire surface, swirl gently to warm and ignite with a stick lighter held just above the surface of the fruit. Allow the flames to extinguish themselves while gently swirling the pan, then immediately remove from heat. Serve over a scoop (or two) of vanilla ice cream. Sit back and enjoy the fruits of your labors!